Coral reefs are diverse ecosystems that are threatened by rising CO2 levels through increases in sea surface temperature and ocean acidification. Here we present a new unified model that links changes in temperature and carbonate chemistry to coral health. Changes in coral health and population are explicitly modelled by linking rates of growth, recovery and calcification to rates of bleaching and temperature-stress-induced mortality. The model is underpinned by four key principles: the Arrhenius equation, thermal specialisation, correlated up- and down-regulation of traits that are consistent with resource allocation trade-offs, and adaption to local environments. These general relationships allow this model to be constructed from a range of experimental and observational data. The performance of the model is assessed against independent data to demonstrate how it can capture the observed response of corals to stress. We also provide new insights into the factors that determine calcification rates and provide a framework based on well-known biological principles to help understand the observed global distribution of calcification rates. Our results suggest that, despite the implicit complexity of the coral reef environment, a simple model based on temperature, carbonate chemistry and different species can give insights into how corals respond to changes in temperature and ocean acidification.